Negative angles (and much more)

[PFuser1232]

How does one prove $\sin{(x+y)} = \sin{x} \cos{y} + \cos{x} \sin{y}$ for all real $x$ and $y$ (not just for $[0, \frac{\pi}{2}]$?
Almost every "geometric proof" I see for formulas like the one I mentioned above, or results such as $\lim_{x \rightarrow 0} \frac{\sin{x}}{x} = 1$, places a restriction (usually $[0,\frac{\pi}{2}]$ on the argument of the sine (or cosine, or tangent) function. The result is then said to be true for all real numbers.
I've seen this over and over again in physics as well. When there is, for example, some force that varies with time, we express the components of that force in terms of the angles given by looking at a convenient case where some right triangle is formed, before applying rules from "classic" geometry such as "the sum of the angles in a triangle is always $\pi$". We then assume the results we get hold true for all angles.
Even when deriving the equations of motion in polar coordinates, while the "analytical derivation" seems to work for angles and scalar components of all magnitudes and signs, the "geometric derivation" almost always involves looking at some convenient situation where the angle/increment in angle is always positive, and where the scalar components of a vector (which can be positive, negative, or zero) are treated as magnitudes.

 

[micromass]

Try to express an arbitrary angle $x$ as an angle in $[0,\pi/2]$ plus some constant.

 

[HallsofIvy]

How does one prove $\sin{(x+y)} = \sin{x} \cos{y} + \cos{x} \sin{y}$ for all real $x$ and $y$ (not just for $[0, \frac{\pi}{2}]$?
Almost every "geometric proof" I see for formulas like the one I mentioned above, or results such as $\lim_{x \rightarrow 0} \frac{\sin{x}}{x} = 1$, places a restriction (usually $[0,\frac{\pi}{2}]$ on the argument of the sine (or cosine, or tangent) function. The result is then said to be true for all real numbers.

The "limit as x goes to 0" can't be said to "be true for all real numbers"!
As for sin(x+ y)= sin(x)cos(y)+ cos(x)sin(y), how that is proved depends on exactly how you have defined the sine and cosine functions. Obviously, if we are defining them for all real numbers we cannot use the trigonometric "opposite side over hypotenuse" definitions. Most textbook use the unit circle: Given a circle with radius one, center at the origin of a xy- coordinate system, if you start at (1, 0) and measure a distance, t, counter-clockwise around the circumference of the circle, cos(t) is the x coordinate of the end point and sin(t) is the y coordinate.

I've seen this over and over again in physics as well. When there is, for example, some force that varies with time, we express the components of that force in terms of the angles given by looking at a convenient case where some right triangle is formed, before applying rules from "classic" geometry such as "the sum of the angles in a triangle is always $\pi$". We then assume the results we get hold true for all angles.

Well, physics texts are notoriously lax about mathematical rigor. However, once again, it does not make sense to talk about "for all angles" in "the sum of the angles in a triangle". Obviously the angles in any triangle must be between 0 and $\pi$. And the proof that the sum of angles is $\pi$ does not require trig functions.

Even when deriving the equations of motion in polar coordinates, while the "analytical derivation" seems to work for angles and scalar components of all magnitudes and signs, the "geometric derivation" almost always involves looking at some convenient situation where the angle/increment in angle is always positive, and where the scalar components of a vector (which can be positive, negative, or zero) are treated as magnitudes.

Then I think you need to look at those proofs more carefully. The proofs I have seen may first treat a "convenient situation" but then extend to other angles.